High Impedance Surface Using A Loop
With Negative Impedance Elements
by
Kyong Hwa Bae
A Thesis Presented in Partial Fulfillment
of the Requirements for the Degree
Master of Science
Approved November 2010 by the
Graduate Supervisory Committee:
James T. Aberle, Chair
Constantine Balanis
George Pan
ARIZONA STATE UNIVERSITY
December 2010
i
ABSTRACT
Antennas are required now to be compact and mobile. Traditional horizontally
polarized antennas are placed in a quarter wave distance from a ground plane
making the antenna system quite bulky. High impedance surfaces are proposed
for an antenna ground in close proximity. A new method to achieve a high
impedance surface is suggested using a metamaterial comprising an infinite
periodic array of conducting loops each of which is loaded with a non-Foster
element. The non-Foster element cancels the loop’s inductance resulting in a
material with high effective permeability. Using this material as a spacer layer, it
is possible to achieve a high impedance surface over a broad bandwidth. The
proposed structure is different from Sievenpiper’s high impedance surface
because it has no need for a capacitive layer. As a result, however, it does not
suppress the propagation of surface wave modes.
The proposed structure is compared to another structure with frequency
selective surface loaded with a non-Foster element on a simple spacer layer. In
particular, the sensitivity of each structure to component tolerances is considered.
The proposed structure shows a high impedance surface over broadband
frequency but is much more sensitive than the frequency selective surface
structure.
ii
TABLE OF CONTENTS
Page
LIST OF FIGURES .............................................................................................iii
CHAPTER
1 INTRODUCTION ........................................................................................... 1
2 ARTIFICIAL MAGNETIC MOLECULES .................................................... 5
2.1 EXTRACTION OF MATERIAL PROPERTIES ...........................7
2.2 DESIGN OF AN ARTIFICIAL MAGNETIC MATERIAL ..........9
2.3 EXTRACTION OF MATERIAL PROPERTIES ...........................13
2.4 CHARACTERITICS OF ARTIFICIAL MAGNETIC
MOLECULES ..................................................................................15
3 FREQUENCY SELECTIVE SURFACES ...................................................... 23
3.1 DESIGN OF FREQUENCY SELECTIVE SURFACES ...............23
3.2 INVESTIGATION OF DISPERSION DIAGRAM ........................26
4 SENSITIVITY INVESTIGATION ................................................................. 27
5 CONCLUSION AND FUTURE WORKS ...................................................... 34
REFERENCE ........................................................................................................ 35
APPENDIX A ....................................................................................................... 36
iii
LIST OF FIGURES
Figure Page
1-1: Sievenpiper’s artificial magnetic conductor and its cross section ................... 2
1-2: An AMM formed by an electrically small loop antenna loaded with passive
circuit elements and its equivalent circuit. .............................................................. 3
1-3: Equivalent transmission model of Sievenpiper’s AMC layers........................ 3
1-4: A theoretical simulation result for a loop loaded with a negative inductor. .... 5
2-1: Definition of AMC bandwidth as the range of frequency for +90 to -90
phase. ...................................................................................................................... 6
2-2: A HFSS simulation setting with material under test in a TEM waveguide..... 8
2-3: Relative permittivity (top) and relative permeability (bottom) of the sample
material for the TEM waveguide in Figure 2-2. ................................................... 11
2-4: An AMM configuration in a TEM waveguide. ............................................. 12
2-5: Relative permeability for AMM realization with 𝑑=-30.5 nH. ............. 12
2-6: AMM relationship between resonant frequency and relative permeability at
low frequencies. .................................................................................................... 13
2-7: An equivalent circuit of the loop with a negative inductance and a negative
capacitance to compensate the Snoek-like limitation. .......................................... 14
2-8: Real and imaginary parts of relative permeability of the loop with and
without of 0.32 pF loaded with 𝑒𝑔 of 37 nH. .................................. 15
2-9: One-port reflection testbed in a waveguide and its close-up in HFSS
simulator. .............................................................................................................. 17
iv
Figure Page
2-10: Relative permeability of the AMM simulated on the one-port reflection
testbed. .................................................................................................................. 18
2-11: Two half loops with the respective 𝑒𝑔 2 and 𝑒𝑔 2. ...................... 19
2-12: Oppositely polarized loop and resulting material characteristics. ............... 20
2-13: Spacer with two-dimensional lattice and its relative permeability. ............. 21
2-14: Two dimensional AMM lattice on PMC testbed. ........................................ 22
3-1: Reflection coefficient phases of an AMM show the bandwidth frequency of
360 MHz. .............................................................................................................. 24
3-2: FSS implementation on a spacer layer. ......................................................... 25
3-3: FSS implemented on the spacer showing the bandgap from 280 MHz to 500
MHz. ..................................................................................................................... 26
4-1: S-parameter magnitude and angle simulated a spacer only model................ 28
4-2: Change of angle of S-parameter by change of inductance for a spacer-only
structure with |Lneg|................................................................................................ 29
4-3: Change of angle of S-parameter by change of inductance for a spacer-only
structure with |Lneg| and| Cpara|. .............................................................................. 29
4-4: A spacer with capacitive pads and the equivalent circuit notation of the
spacer itself and the combined structure. .............................................................. 30
4-5: A FSS pads combined with non-Foster elements and its equivalent circuit. 31
4-6: Magnitude and Angle of S-parameter from FSS structure controlled by non-
Foster elements on its layer................................................................................... 32
v
Figure Page
4-7: Sensitivity of Angle of S11 with respect to change of negative inductance on
FSS pads................................................................................................................ 33
1
1 INTRODUCTION
In the past two decades, much electric equipment has been asked to be
compact and mobile operating in a broad range of frequencies. Recent studies
have looked at an integrating antenna and its ground in closer proximity.
Typically, antennas are placed on a perfect electric conductor (PEC) at a quarter
wave distance to maximize their efficiency. For example, a dipole antenna
operating at 350 MHz should be placed in a distance of about 21 cm (or about
8.44 inches) placed horizontally from a ground plane. In many applications, the
antenna must be placed much closer to the ground plane. While it can be a good
reflector, it is not a good ground since the phase of the incident wave is reversed
according to the image theorem [1]. It cancels the phase of incident and reflected
waves and shows low radiation efficiency. This constraint of the antenna system
has been an obstacle to realize a low-profile and compact system.
A photonic bandgap (PBG) structure, also known as an artificial magnetic
conductor (AMC), can been considered as a ground plane for low profile antennas
in order to reduce the distance between an antenna and its ground by filling the
spacer with an artificial medium. The height, h, can be calculated by
h = 𝑐0
𝑓𝑟𝑒𝑠𝑜𝑛𝑎𝑛𝑡∙1
4
(1.1)
In the past decades, the designs of periodic structures have been developed to
control the propagation characteristics of electromagnetic fields [2]. They enable
the development of materials that exhibit novel electromagnetic properties which
2
are not available in nature and can be easily optimized for the desired application.
These materials are known as metamaterials [3]. In 1999, Sievenpiper and
Yablonovitch first introduced a high impedance surface for the microwave and
antenna domains [4]. They suggested mushroom-type surfaces which are similar
to corrugated surfaces except for the fact that they exhibit two-dimensional
periodicity in order to prevent the propagation of both vertically and horizontally
polarized waves along its surface [5].
In the following chapters, a customized artificial magnetic conductor (AMC)
for ultrahigh frequencies (UHF) is suggested that consists of a spacer, a frequency
selective surface (FSS), and a ground surface as shown in Figure 1-1. The AMC
can be modeled as shown in Figure 1-2 for normal wave incidence by substituting
Figure 1-1: Sievenpiper’s artificial magnetic conductor and its cross section
3
a transmission line for the spacer and a capacitor for the FSS. Because the
bandwidth is related to the permeability and the thickness of the spacer as
∆𝑓
𝑓𝑟𝑒𝑠𝑜𝑛𝑎𝑛𝑡= 2𝜋𝜇𝑟
𝜆0 (1.2)
the spacer should be designed to get a desired permeability [6]. A short
explanation is in Appendix 1. When the thickness is assumed to be 1.7 inches and
the required frequency range is assumed to be 200 MHz to 500 MHz, for UHF
application, the relative permeability should be greater than 2.7.
The high permeability spacer is realized by embedding electrically small
artificial magnetic molecules (AMMs) in a host medium. The AMM is formed by
electrically small loop antennas loaded with passive electrical circuit elements [7].
The loop can be represented by the equivalent circuit shown in Figure 1-3. The
Figure 1-3: Equivalent transmission model of Sievenpiper’s AMC layers.
Figure 1-2: An AMM formed by an electrically small loop antenna loaded with
passive circuit elements and its equivalent circuit.
4
magnetic polarizability of the artificial molecule can be expressed as
α =𝑚
𝐻=
−𝑗𝜔𝜇0 4
𝑅𝑙𝑜𝑠𝑠 + 𝑗𝜔 𝑙𝑜𝑜𝑝 + 𝑍𝐿 (1.3)
Then, the relative permeability of the material is
= 1 + (1.4)
Here, it is assumed that the coupling interactions between molecules are ignored,
and the host medium has the same relative permeability as that of free space. To
cancel the inductance of the loop inductance, 𝑙𝑜𝑜𝑝 , the passive load must be
− 𝑙𝑜𝑎 , which is a negative inductance. When 𝑙𝑜𝑎 is slightly higher than
𝑙𝑜𝑜𝑝 , the relative permeability will be increased greatly.
A simple circuit simulation result is shown in Figure 1-4. When the negative
inductance is controlled as desired, the relative permeability is also controlled.
The next chapters will show how to take out the characteristics of the loop and to
cancel its inductance in a unit cell, which is a part of a periodic structure.
5
Figure 1-4: A theoretical simulation result for a loop loaded with a negative inductor.
0 1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
16
18
Frequency, GHz
Rel
ativ
e P
erm
eabil
ity
imag
real
a
W
ZL= -jLd
0 1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
16
18
Frequency, GHz
Rel
ativ
e P
erm
eabil
ity
imag
real
a
W
ZL= -jLd
Circuit Theory Model
6
2 ARTIFICIAL MAGNETIC MOLECULES
An effective low-profile ground plane for an antenna requires certain
characteristics to support the effective radiation of the antenna with reduction of
undesired back lobes and reactive coupling to nearby circuits. The structure can
be a high impedance photonic bandgap (PBG), also known as an artificial
magnetic conductor (AMC). A designed AMC helps to realize a planar low
profile antenna. For a bandwidth between +90 and -90 phases shown in Figure
2-1, the frequency bandwidth should be at least 300 MHz with a center frequency
of 350 MHz. According to Equation (1.2), a spacer layer thickness of 1.7 inches
must have the permeability of the spacer greater than 2.7. An artificial material is
designed with an effective permeability of about 3.6 to achieve broad bandwidth.
Figure 2-1: Definition of AMC bandwidth as the range of frequency for +90 to
-90 phase.
7
2.1 EXTRACTION OF MATERIAL PROPERTIES
To achieve the desired bandwidth in a given form factor, we decided to make
a spacer with a relative permeability of 3.6. To verify the characteristics of the
material, a procedure should be established to extract the material properties. The
material is simulated in Ansoft HFSS (High Frequency Structure Simulator) for
three-dimensional full-wave electromagnetic (EM) simulations. The material to
be investigated is placed in the middle of a waveguide which has two excitation
ports on both end sides as shown in Figure 2-2. The waveguide is set with one
pair of opposing sides with perfect electric conductor (PEC) walls and the other
pair with perfect magnetic conductor (PMC) walls to realize a transverse
electromagnetic (TEM) waveguide. From image theory this is equivalent to an
infinite periodic structure in the x-y plane. Note that PMC boundaries can be
replaced by symmetry boundaries in the simulator.
The procedures to numerically evaluate the effective medium properties of a
given material are developed. The material S-parameters are extracted from HFSS,
exported, and post-processed in Mathworks MATLAB.
First, the 2 port TEM waveguide containing a material sample is simulated in
HFSS and the S-parameters are extracted. Then, the data is calculated in
Mathworks MATLAB. The reference plane is shifted so that the data is only for
the material under test (MUT).
[ ] = [𝑅][ ][𝑅] (2.1)
where
8
[ ] is S-parameters from HFSS analysis,
[𝑅] = ( ) ,
= 0( − 𝑑)
= overall length of TEM waveguide, and
d = thickness of material sample.
The parameters are then converted to ABCD parameters [8].
A =(1 + 11)(1 − 22) + 12 21
2 21
B = 𝑍0(1 + 11)(1 − 22) − 12 21
2 21
C =1
𝑍0
(1 − 11)(1 − 22) − 12 212 21
D =(1 − 11)(1 + 22) + 12 21
2 21
(2.2)
Then, the material’s propagation constant () and characteristic impedance (Zc)
of equivalent transmission line are calculated as
MUTMUTx
y
z
Figure 2-2: A HFSS simulation setting with material under test in a TEM
waveguide.
9
1arccos
2
A Dj
d
and C
BZ
C . (2.3)
Finally, the material properties are evaluated using
0 0
Cr
Zj
k
and 0
0
r
C
jk Z
(2.4)
where k0 is the propagation constant and 0 is the wave impedance of free space.
To verify the validity of the approach, some known test materials are
examined. For instance, a 4-cm slab of a lossy magneto-dielectric material with its
relative permittivity r=2.2-j2.2 and relative permeability µ r=3.6-j3.6, is placed in
a 50-cm long TEM waveguide. The post-processed results are shown in Figure 2-
3. Notice that both graphs closely match both the real and imaginary parts of
relative permittivity and relative permeability.
2.2 DESIGN OF AN ARTIFICIAL MAGNETIC MATERIAL
The characteristics of any material in the TEM waveguide are extracted using
the approach described in the previous section. A proposed artificial magnetic
molecule (AMM) in the previous chapter consists of a loop loaded with a negative
inductance, 𝑙𝑜𝑎 . The value of 𝑙𝑜𝑎 is initially approximated using an
equivalent circuit of a square loop [1]. 𝑙𝑜𝑜𝑝 in Figure 1-3 is estimated as
following:
𝑙𝑜𝑜𝑝 = 2𝜇0
𝜋[ln
𝑏− 0 774] (2.5)
10
where a is a side length of a square loop and b is a wire radius. For a loop with a
side length of 1.2 inches, the other side length of 0.5 inches, and the wire width of
75 mil, a is assumed to be 0.85 inches and b is 37.5 mil. Then 𝑙𝑜𝑜𝑝 is calculated
roughly to be 40 nH. With this assumption, a loop loaded with a negative element
is implemented inside a waveguide in HFSS as shown in Figure 2-4.
A negative inductance, 𝑙𝑜𝑎 , assumed to be slightly larger than 𝑙𝑜𝑜𝑝 , is
placed inside the loop since it is a frequency domain solver. HFSS allows
inputting negative lumped element numbers even though the negative impedance
is a non-Foster element. Since the loop inductance was roughly calculated, the
value of 𝑙𝑜𝑎 is determined by a trial-and-error procedure.
Figure 2-5 shows a computed result simulated with 𝑙𝑜𝑎 of -30.5nH. A
desired relative permeability was achieved at a lower frequency than its resonant
frequency. Since the design is targeted to have a broadband application, the
resonant frequency should be moved to a higher frequency. However, the
simulations showed that a higher resonant frequency has lower permeability at
low frequencies. Those two relations are plotted in a coordinate shown in Figure
2-6. This relation is similar to Sneok’s limit, which is a relationship between
frequency of maximum absorption and permeability at low frequency [9].
11
Figure 2-3: Relative permittivity (top) and relative permeability (bottom)
of the sample material for the TEM waveguide in Figure 2-2.
12
Figure 2-5: Relative permeability for AMM realization with 𝐿𝑙𝑜𝑎𝑑=-30.5 nH.
Figure 2-4: An AMM configuration in a TEM waveguide.
13
2.3 EXTRACTION OF MATERIAL PROPERTIES
J. L. Snoek first stated Snoek’s limit, which is a rule for magnetic materials.
The higher the permeability, the lower the frequency at which absorption and
dispersion set in; the product of the resonant frequency and the initial
permeability is constant.
The designed spacer also exhibits the Snoek-like characteristic feature. The
product of the resonance frequency and the DC (low frequency) permeability is
constant as shown in Figure 2-6, which is reminiscent of Snoek’s limit [9]-[11].
A product value is desired to be higher and the relationship between the resonant
frequency and the relative permeability at low frequency is plotted as in the right
Figure 2-6: AMM relationship between resonant frequency and relative
permeability at low frequencies.
14
top corner in the coordinate as possible. Variations in the geometric dimensions
cannot overcome this limitation. It can be considered that resonance is a result of
the parasitic capacitance mode of the loop itself.
The parasitic capacitance of the loop can be compensated by a negative
capacitance as seen in Figure 2-7. The loop with the cancelled parasitic
capacitance exhibits a high relative permeability over a broader frequency range.
An example of the achievement is shown in Figure 2-8 with 𝑛𝑒 of 37 nH and
𝑝𝑎𝑟𝑎 of 0.32 pF. The presence of the negative capacitance causes the resonant
frequency to move higher than the frequency of interest. The relative permeability
exhibits close to a constant value now.
Figure 2-7: An equivalent circuit of the loop with a negative inductance and a
negative capacitance to compensate the Snoek-like limitation.
15
2.4 CHARACTERITICS OF ARTIFICIAL MAGNETIC MOLECULES
The artificial magnetic molecule is designed to achieve a relative
permeability of about 3.6 in the UHF bandwidth from a loop loaded with negative
impedance. The loop in the middle of a TEM waveguide is analyzed in the
reflection testbed as shown in Figure 2-9. A waveport for exciting an incident
wave is located on the opposing plane and collects the S-parameter information
from the reflection coefficients. With the change of HFSS environment settings,
the calculation in MATLAB is also changed for one-port reflection calculation.
Then, a constant relative permeability is achieved as seen in Figure 2-10.
Figure 2-8: Real and imaginary parts of relative permeability of the loop with
and without 𝐶𝑝𝑎𝑟𝑎 of 0.32 pF loaded with 𝐿𝑛𝑒𝑔 of 37 nH.
16
The AMM-AMC comprising loops loaded with negative impedances have
been successfully simulated. This AMC will exhibit only TE surface wave mode
suppression because it has no vias. The demonstrations show featured spacer
layers with AMMs arranged only in a one-dimensional lattice. Such a structure
has AMC properties that exist only for one cardinal polarization of the incident
wave, which is the one whose magnetic field interacts with the loops. An
extension of this structure has also been successfully simulated. It has a two-
dimensional AMM lattice allowing for interaction with waves polarized in either
a cardinal direction or indeed polarized in an arbitrary direction, since any
arbitrary polarization is a superposition of these two polarizations for a normally
incident plane wave.
Before placing two crossed substrates in the reflection testbed simulation, a
split loop in a single plane is verified. In the simulation the negative impedance
consisting of a negative inductor and a negative capacitor in parallel is split in half.
The values of each inductor and capacitor are respectively half and twice as large
as the previously simulated loop. Thus, | 𝑛𝑒 | = 37 n 2 and | 𝑛𝑒 | =
0 32 2 and they are placed as shown in Figure 2-11.
The two-dimensional AMM lattice geometry and simulation results are
shown in Figure 2-11. Note that the negative impedance elements are tuned to
| 𝑛𝑒 | = 40
2 and | 𝑛𝑒 | = 0 32 2 to achieve the desired properties.
The simulated results show that the spacer layer exhibits more or less
constant permeability over UHF frequencies.
17
Figure 2-9: One-port reflection testbed in a waveguide and its close-up in HFSS
simulator.
18
To evaluate the effective permeability of the spacer layer, the reflection
testbed is terminated by a PEC ground plane. To evaluate the effective
permittivity, the ground plane is changed from PEC to PMC as shown in Figure
2-12. Both results agree very well at the lower frequencies and diverge at the
higher frequencies. It is expected that the results of the two-port testbed
simulation are more accurate.
The designed UHF AMM-AMC has a relative permittivity of 1.3+j0 and a
relative permeability of 3.6+j0.
100 200 300 400 500 600 700 800 900 10000
2
4
6
8
10
12
14
16
18
20
freq(MHz)
real(
r)
Figure 2-10: Relative permeability of the AMM simulated on the one-port
reflection testbed.
19
100 200 300 400 500 600 700 800 900 10000
2
4
6
8
10
12
14
16
18
20
freq(MHz)
real(
r)
Figure 2-11: Two half loops with the respective |𝐿𝑛𝑒𝑔| 2 and |𝐶𝑛𝑒𝑔| 2.
20
Figure 2-12: Oppositely polarized loop and resulting material characteristics.
100 200 300 400 500 600 700 800 900 10000
2
4
6
8
10
12
14
16
18
20
freq(MHz)
real(
r)
100 200 300 400 500 600 700 800 900 1000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
freq(MHz)
imag(
r)
PMC
21
Figure 2-13: Spacer with two-dimensional lattice and its relative permeability.
100 200 300 400 500 600 700 800 900 10000
2
4
6
8
10
12
14
16
18
20
freq(MHz)
real(
r)
100 200 300 400 500 600 700 800 900 1000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
freq(MHz)
imag(
r)
2-port
1-port
22
Figure 2-14: Two dimensional AMM lattice on PMC testbed.
100 200 300 400 500 600 700 800 900 10000
2
4
6
8
10
12
14
16
18
20
freq(MHz)
real(
r)
2-port
1-port
100 200 300 400 500 600 700 800 900 1000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
freq(MHz)
imag(
r)
2-port
1-port
23
3 FREQUENCY SELECTIVE SURFACES
The design of a spacer layer for an artificial magnetic conductor (AMC) has
been discussed in the previous chapter. To provide anti-resonance at the
frequency of interest, a capacitive layer is usually added on a spacer layer as a
Sievenpiper’s model. Therefore, the structure will emulate high-impedance
condition, experienced by an antenna placed one quarter wavelength above a PEC.
The design of a capacitive layer is based on the concept of a frequency
selective surface (FSS). In some sense, this terminology is appropriate in the
context of AMCs since the capacitive layer design will also influence the surface-
wave suppression properties of an AMC.
In this chapter, we design a FSS to obtain a desired reflection coefficient
phase behavior in an AMC, which is incorporated with the previously designed
spacer layer. Then, we analyze the surface wave properties of the resulting AMC
using the effective media model implemented in MATLAB and the structural
simulation in HFSS.
3.1 DESIGN OF FREQUENCY SELECTIVE SURFACES
The AMC is modeled as a two layer bi-uniaxial model as shown in Figure A-1.
The spacer is represented as Y1, the FSS as Y2, and the radiation space as Y3. The
reflection coefficient, , is derived for both polarizations of the incident wave, but
usually a normal angle of incidence is assumed, 𝑛 = 0 . For example, the
24
reflection coefficient for the TM mode is shown in Equation (3.1) based on the
definitions of Equation (A.6)-(A.8).
= 3 − 3 +
(3.1)
where
= 2 𝐿 + 2t n (𝛾2𝑑2)
2 + 𝐿t n (𝛾2𝑑2) and
𝐿 = 1 th (𝛾1𝑐 1𝑑1).
The magnitude and the phase of the reflection coefficient for a normal
incidence are shown in Figure 3-1. Notice that each admittance, Y, is calculated
as in Appendix A. With the obtained relative permittivity and relative
Figure 3-1: Reflection coefficient phases of an AMM show the bandwidth
frequency of 360 MHz.
25
permeability of the spacer from the previous chapter, an initial design can be
obtained using the effective media model in MATLAB. Based on this MATLAB
model, the geometry was simulated in HFSS as seen in Figure 3-2.
It was needed to adjust the geometry several times in order to obtain the
desired reflection coefficient behavior in the HFSS simulations. We attribute this
need for iteration to inaccuracies in the calculation of the FSS layer capacitance in
the MATLAB model.
Figure 3-2: FSS implementation on a spacer layer.
26
3.2 INVESTIGATION OF DISPERSION DIAGRAM
The AMC design discussed above exhibits only TE surface wave mode
suppression. To suppress TM surface waves, we need to add properly positioned
vertical pins (vias) to the design. Using MATLAB and the dimensions that
correspond to Figure 3-1 results, the dispersion diagram is plotted in Figure 3-3.
Since the TE bandedge is already fixed with the FSS dimension, only the TM
bandedge can be adjusted. In addition, the AMC should be appropriately designed
in order to suppress all surface waves as shown in Figure 3-3.
0 50 100 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x (rad/m)
Fre
quency (
GH
z)
Dispersion Diagram for the designed AMC, Pvia=1.3times Pfss
TM
TM
TM
TE
TE
TE
light
Figure 3-3: FSS implemented on the spacer showing the bandgap from
280 MHz to 500 MHz.
27
4 SENSITIVITY INVESTIGATION
As discussed in Chapter 2, a high impedance surface is achieved over a broad
bandwidth without a FSS layer and with a FSS layer. Because their sensitivities to
non-Foster elements are different, S-parameter’s sensitivities to the changes of
negative lumped elements are investigated.
For a spacer layer only structure as shown in Figure 2-9, the negative
inductance is changed to see how much it affects its S-parameter. With | 𝑛𝑒 | of
-44 nH, the magnitude and angle of S11 are shown in Figure 4-1. It shows the
center frequency of 300 MHz and its bandwidth of 200 MHz.
First, a spacer loaded with a negative inductance is investigated without the
Snoek-limit consideration. The angle of S11 at the frequency of 300 MHz, the
initial center frequency, is plotted as shown in Figure 4-2. As the inductance is
changed by -0.5 % to 1.3 %, the angle of S11 experiences changes of +30 degrees
to -30 degrees. The rate of the change is expressed by
𝑛𝑒 −80 − 65
n (4.1)
Since the loop shows the Snoek-like limitation, the spacer should have
negative capacitance to cancel the loop parasitic capacitance. When the non-
Foster capacitance is included, the sensitivity rate is sharply increased by
𝑛𝑒 −220
n (4.2)
as shown in Figure 4-3. The sensitivity of the structure is also compared with a
sensitivity of the structure with a FSS layer.
28
Figure 4-1: S-parameter magnitude and angle simulated a spacer only
model.
100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
freq(MHz)
Mag(S
)
100 200 300 400 500 600 700 800 900 1000
-150
-100
-50
0
50
100
150
freq(MHz)
angle
(S)
29
Figure 4-2: Change of angle of S-parameter by change of inductance for a
spacer-only structure with |Lneg|.
-200
-150
-100
-50
0
50
100
150
200
-80 -60 -40 -20 0
Lneg
An
gle
of
S1
1
Figure 4-3: Change of angle of S-parameter by change of inductance for a
spacer-only structure with |Lneg| and| Cpara|.
y = -219.18x - 8703-200
-150
-100
-50
0
50
100
150
200
-70 -60 -50 -40 -30 -20 -10 0
Lneg
An
gle
of
S11
30
The property of a simple spacer layer is first extracted and a FSS structure
with negative elements is combined as shown in Figure 4-4. Then, the spacer
capacitance is cancelled out using non-Foster elements on the FSS as shown in
Figure 4-5. The S-parameters are shown in Figure 4-6.
Now, the sensitivity of the angle of S11 is investigated with respect to change
of negative inductance. The relationship is plotted in Figure 4-7.
Figure 4-4: A spacer with capacitive pads and the equivalent circuit notation of
the spacer itself and the combined structure.
31
Figure 4-5: A FSS pads combined with non-Foster elements and its equivalent
circuit.
32
Figure 4-6: Magnitude and Angle of S-parameter from FSS structure controlled
by non-Foster elements on its layer.
100 200 300 400 500 600 700 800 900 10000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
freq(MHz)
Mag(S
)
100 200 300 400 500 600 700 800 900 1000
-150
-100
-50
0
50
100
150
freq(MHz)
angle
(S)
33
The change of the angle of S11 at the center frequency of 300 MHz is
observed and its relationship is shown as
𝑛𝑒 −7 8 − 7 6
n (4.3)
It shows that the angle of S11 varies from 30 deg to +30 deg as |Lneg| changes by -5%
~ 7%. That shows the FSS layer with non-Foster elements will be less sensitive
than the spacer-only layer with the loop loaded non-Foster elements on it.
Figure 4-7: Sensitivity of Angle of S11 with respect to change of negative
inductance on FSS pads.
34
5 CONCLUSION AND FUTURE WORKS
An AMC structure implementation for a high impedance surface is interesting
and challenging. An AMC is suggested with loops loaded with non-Foster
elements on them. The structure has a spacer layer only without a FSS layer,
unlike Sievenpiper’s structure. The spacer only structure is much simpler even
though it does not suppress the propagation of surface wave mode and it is
sensitive with regard to non-Foster elements.
The structure is also compared to a FSS layer only model. Regarding the
sensitivity, a FSS layer is expected to reduce the sensitivity. The structure can be
implemented to combine the spacer layer and the FSS layer together to make it
less sensitive.
When a negative impedance converter with non-Foster elements is available
for the frequency of interest, it will provide the Lneg and Cneg for the AMM loops
or FSS layers. This will enable a low-profile UHF antenna less than /4 with a
high impedance ground surface.
35
REFERENCE
[1] Balanis, C. A., Antenna Theory Analysis and Design, John Wiley and Sons,
Inc., 2005.
[2] Kildal, P., Kishk, A. and Maci, S. 1, “Special issue on artificial magnetic
conductors, soft/hard surfaces, and other complex surfaces,” IEEE Trans.
Antennas Propagat., Vol. 53, pp. 2-7, 2005.
[3] Ziolkowski, R. W. and Engheta, “Matamaterial special issue introduction,”
IEEE Trans. Antennas Propagat., Vol. 51, No. 10, pp. 2546-2549, 2003.
[4] Sievenpiper, D., Zhang, L., Broas, R. F. J., Alexopolous, N. G., and
Yablonovitch, E., “High-impedance electromagnetic surface with a forbidden
frequency ban,” IEEE Trans. Microwave Theory Tech., Vol. 47, No. 11, pp.
2059-2074, 1999.
[5] Sievenpiper, D. F., “High-impedance electromagnetic surfaces,” Ph. D.
dissertation, University of California Los Angeles, 1999.
[6] Sanchez, V. C., McKinzie III, W. E., and Diaz, R. E., “Broadband antennas
over electronically reconfigurable artificial magnetic conductor surfaces,”
Proceedings of Antenna Applications Symposium, Vol. 1, pp. 70-83, 2001.
[7] Ziolkowski, R. W. and Auzanneau, F., “Artificial molecule realization of a
magnetic wall,” J. Appl. Phys, Vol. 82(7), pp. 3192-3194, 1997 .
[8] Pozar, David M., Microwave Engineering, John Wiley and Sons, Inc., 2005.
[9] Snoek, J. L., “Dispersion and absorption in magnetic ferrites at frequencies
above one mc/s,” Physica, Vol. 14(4), pp. 207-217, 1948.
[10] Jankovskis, J., “Empirical relations analogous to Snoek's law for account of
poly-crystalline ferrites grain size effects,” Scientific Proceedings of Riga
Technical University in series Telecommunications and Electronics, Vol. 2, pp.
68- 77, 2002.
[11] Snoek, J. L., New development in ferromagnetic materials, 1966.
[12] Clavijo, S., Diaz, R. E. and McKinzie, W. E., “Design methodology for
Sievenpiper high-impedance surfaces: An artifical magnetic conductor for
positive gain electrically small antennas,” IEEE trans. Antennas Propagat., Vol.
51, No. 10, pp. 2678-2690, 2003.
36
APPENDIX A
IMPEDANCE CALCULATION OF A BI-UNIAXIAL STRUCTURE
FOR TM MODE
37
The impedance of a two-layer bi-uniaxial structure is calculated for the TM
mode. Each layer can be modeled as a transmission line. The entire configuration
is shorted at the one end as shown in Figure A-1.
For the bi-uniaxial structure, the material characteristics of both layers are
applied to Equation (A.6) - (A.8).
The second layer is corresponds to the capacitance instead of a FSS as shown
in Figure 1-3. The admittance of this layer is designated by Y2.
2𝑡 = 2 = 2 = 2
𝑗𝜔 0 2𝑛 = 2 = 1 (A.1)
2𝑡 = 2 = 2 = 1 2𝑛 = 2 = 2
𝑗𝜔 0 (A.2)
The first layer is a periodic structure where a unit cell has a via. It is a spacer
region which has the medium properties derived in Section 2.4. The material
properties are calculated as a Brown’s rodlike medium shown in [12].
Figure A- 1: Equivalent transmission line representation of two layer bi-uniaxial
structure.
38
1𝑡 = 1 = 1 = 𝑟 𝑠𝑝𝑎 𝑒𝑟1 −
1 + (A.3)
1𝑛 = 1 = 𝑟 𝑠𝑝𝑎 𝑒𝑟 −𝑐2
𝜔2𝜇𝑟 4𝜋 2( − − 1)
(A.4)
1 = 1 = 𝑟 𝑠𝑝𝑎 𝑒𝑟
2𝑛 𝑟 𝑠𝑝𝑎 𝑒𝑟 and 1 = (1 − )𝜇𝑟 (A.5)
where
α =𝜋 𝑎
2
2
𝑎 = ,
And = l n th
From these material properties, each of admittance can be calculated. Here,
TM wave propagation is assumed, and the incident wave impinges at an angle of
with respect to the normal direction.
Figure A- 2: Unit cell of via array [12].
39
1 = 0 1𝑡γ1𝑐 1
(A.6)
2 = 0 2𝑡γ2𝑐 2
(A.7)
3 = 0
γ0𝑐 𝑛 (A.8)
Where
inc = incident angle,
t, n = transverse and normal directional permittivity,
µ t, µn = transverse and normal directional permeability,
1 = t 1 (
1𝑡𝜇1𝑡 n2 𝑛
− 1𝑡 1𝑛)
2 = t 1 (
2𝑡𝜇2𝑡 n2 𝑛
− 2𝑡 2𝑛)
γ1 = √− 02
1𝑡𝜇1𝑡 1𝑛 1𝑡 n2 1 + 1𝑛 2 1
γ2 = √− 02
2𝑡𝜇2𝑡 2𝑛 2𝑡 n2 2 + 2𝑛 2 2
